2 nuria calvet and paola d alessio

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1 2 nuria calvet and paola d alessio PROTOPLANETARY DISK STRUCTURE AND EVOLUTION. Introduction The standard picture of star formation, based on several decades of observations and models, can be summarized as follows. Stars are born in dense cores inside molecular clouds, which are the densest regions of the interstellar medium (ISM). These dense cores collapse under their self-gravity, and because they have some angular momentum, they end up forming disklike structures and/or multiple stellar systems. When a single star s disk is formed, the lowest-angular-momentum core material falls toward its center, where the star builds up. The rest of the material falls onto the circumstellar disk, which surrounds the young star during its first few million years of life. These young disks are called accretion disks because their central star acquires mass from them. Disks evolve during their lifetime. Viscosity and gravitational torques transfer angular momentum to a fraction of the disk material that ends up moving toward the outer regions, increasing the disk radius. On the other hand, the fraction of material that has lost angular momentum falls toward the star, increasing the stellar mass. Disk mass is replenished by the molecular dense core until the latter is dissipated by a disk wind or some other mechanisms. Gravitational torques must be important in transferring angular momentum when a disk has a large mass compared with its central star (see the chapter by Durisen, this volume). Simultaneously, dust grains grow inside the disk, from micrometer sizes like grains found in the diffuse interstellar medium to millimeter, meter, and even kilometer sizes, eventually to form planetesimals. The disk is a natural place to grow a planetary system if it happens to last long enough. However, how long a disk should live to actually form planets depends on the details of the planet-formation process, which are a matter of debate. Recently, timescales have been observationally constrained, helping _ch2_P.tex /28/2 6:8 page 4

2 PROTOPLANETARY DISKS / 5 define the finer details of how a planetary system is built up. What seems to be clear, on the basis of the number of extrasolar planets found so far, is that the formation of planetary systems around stars is not a rare process. This is why accretion disks around young stars are frequently called protoplanetary disks, reflecting their potential to form planets. The final destiny of the disk mass is to be part of planets, to be accreted by the star, or to be lost in a photodissociated wind or by a stellar encounter. In this chapter we will describe what has been learned about disk structure and evolution from the combination of observations and modeling. In 2 we give a summary of the observations on which models rely. In 3welist evidence for accretion; review magnetospheric accretion, the current paradigm for accretion; an describe determinations of accretion luminosity and massaccretion rate, concentrating in low-mass Young Stellar Objects YSOs In 4 we describe the physics of irradiated accretion disks and how it relates to the properties of solids in the disk and to accretion. Finally, in 5, we describe the effects of dust evolution in disks and evidence that the expected evolution is taking place. 2. Observational Overview Observations give us quantitative information about the reality we are able to measure. This makes them the basis of any modeling effort, not only by motivating the generation of new sets of models but also by constraining models, verifying their value as an appropriated description of reality. Frequently, one is able to explain a particular observation with many models, but only a few give a consistent picture of the whole set of observations available and allow predictions that guide new observations for further tests. In this section we summarize observations of young stars, related to the disk-model interpretation. This is not a review of all the observations involving young stars, but only an attempt to introduce some basic observational information and to illustrate its importance for constructing the models described in the following sections. When a protoplanetary disk model is calculated, the importance of the central star is sometimes underestimated. The star is the ultimate source of the disk energy, heating it by irradiation and producing the gravitational field where the disk mass falls, releasing energy during the accretion process. Thus the properties of the central stars are an essential input for modeling disks. Figure 2. shows the location in the Hertzsprung-Russell (HR) diagram of the most commonly studied visible YSOs, together with evolutionary tracks _ch2_P.tex /28/2 6:8 page 5

3 6 / NURIA CALVET AND PAOLA D ALESSIO Fig. 2.. Location of young stellar objects in HR diagram. Shown are substellar T Tauri stars (YBDs), classical T Tauri stars (CTTSs), weak-line T Tauri stars (WTTSs), Herbig Ae stars and their predecessors, and intermediate-mass T Tauri stars (IMTTSs). Sources for the data: substellar objects [5], Tauru s CTTSs and WTTSs [4], and Herbig Ae [95]. Zero-age main sequence (ZAMS, long-dashed line), evolutionary tracks (solid lines, labeled by the corresponding stellar mass in solar masses), and isochrones (dashed lines, corresponding to.3,, 3,, and 3 Myr from top to bottom) from [6] and [3]. and isochrones from [6] and [3]. These objects are given different names depending on their mass: Herbig Ae/Be stars (HAeBe), A and B stars with emission lines [9], with masses M < M < 8M. T Tauri stars (TTSs):, characterized by late-type spectra superimposed by strong emission lines [9], with masses.8 < M M ; Young brown dwarfs (YBDs), substellar objects with masses M <.8 M that will never reach a central temperature high enough to burn hydrogen. As shown in Fig. 2., TTSs and YBDs tend to be on the Hayashi track, while HAeBe are much closer to the main sequence, and all have ages Myr. However, note that some TTSs have masses comparable to the HAeBe (the intermediate-mass TTSs, IMTTSs) and will end up as HAeBe as they evolve along the radiative tracks. Young stars are subject to different classification schemes, based on observational criteria. These categories are usually related to evolutionary stages, but we will see that the picture is not as clear as it seems at first sight. For instance, according to the slope dlog(λf λ )/dlogλ of their spectral energy distribution (SED) in the 2.2 to 25 μm range [7, 6], YSOs have been classified as _ch2_P.tex /28/2 6:8 page 6

4 PROTOPLANETARY DISKS / 7 Class I, with positive slope. These are thought to be still surrounded by infalling material from which they form. Class II, with negative slope, but still flatter than the Rayleigh-Jeans slope expected for stellar photospheres, λf λ λ 3. Their excess is explained as produced by the disk. Class III, with photospheric slopes [7]. They appear to have dissipated their disks. This classification scheme, which is based on the slope of the SED, is applicable to YSOs of all masses. Another classification scheme, applied only to T Tauri stars, is based on the strength of their emission lines. According to this scheme, TTSs can be classified as follows: Classical TTSs (CTTSs), with an equivalent width of Hα larger than Å [92]. These stars also show strong excesses in line and continua above the intrinsically photospheric fluxes and are thought to be accreting mass from the disk (see 3). Weak-line TTSs (WTTSs), with Hα equivalent widths lower than this limit [92]. More recent studies indicate that the Hα equivalent width separating both classes depends on spectral type [79]. These stars are thought not to be accreting mass from the disk. The identification of CTTSs with Class II objects and WTTSs with Class III objects is usually made under the assumption that the disk that produces the near-ir emission characterizing the Class II objects is an accretion disk. In this case, the potential energy released by the accretion process is responsible for the excess in line and continuum emission seen in CTTSs in the optical and shorter wavelengths. However, observations of the Spitzer Space Telescope in the last 5 years have altered this simple picture. Although most CTTSs are Class II, some WTTSs may show emission from remnant disks; also, some stars with clear signs of being accreting may show no excess in the μm range [35] and thus could not be classified as Class II. Thus an easy one-toone correspondence cannot be made in all cases, and all indicators have to be examined to determine the physical properties of a given object. Another inference usually drawn from these classifications is that because WTTSs have already lost their disks, they should be older than CTTSs; however, examination of Fig. 2. shows that CTTSs and WTTSs coexist in the HR diagram; that is, they have similar ages, indicating that age is not the only factor in determining disk dissipation. The accretion-based classification of CTTSs and WTTSs does not extend to the HAeBe. These stars are selected because they have emission lines in their _ch2_P.tex /28/2 6:8 page 7

5 8 / NURIA CALVET AND PAOLA D ALESSIO spectra, while most stars in the corresponding spectral range show absorption lines only. Moreover, they have infrared excesses consistent with the presence of disks [97]. This means that all are Class II objects and, if anything, would be intermediate-mass analogs of the CTTS. Thus a classification scheme often used in the literature for these objects is based on the shape of their infrared SED [23], and the differences are thought to be due to different properties in the disk: Group I, objects with a continuum that can be reconstructed by a power law and a blackbody. Members of the subgroup Ia have solid-state bands present in their SEDs, and those of Ib have no solid-state bands. Group II, objects with a continuum that can be reconstructed by only a power law. Again, members of subgroup IIa have solid-state bands, and those of IIb have no solid-state bands in their spectra. The large midinfrared continuum excess observed in SEDs of HAeBe in group I is usually associated with the emission of flared disks that are heated by stellar irradiation. The smaller excess shown by stars in Group II is associated with a flat outer disk in the shadow of a puffed-up inner region [23, 52] or with the settling of dust grains [55]. CTTSs and WTTSs show emission lines in their spectra, but the lines are much broader and generally stronger in CTTSs. In addition, CTTS show veiling of their photospheric absorption lines [79]; i.e., these lines are less deep than those of main-sequence stars of the same spectral type. This is interpreted in terms of an excess in continuum that adds to the intrinsic photospheric flux. Veiling is measured in terms of the veiling parameter r λ = F v /F ph, where F v is the excess flux and F ph is the photospheric flux at the same wavelength. The veiling parameter increases as wavelength decreases [75, 8]; the excess flux dominates in the ultraviolet (UV) and shorter wavelengths because the intrinsic photospheric flux drops [76]. The excess luminosity in CTTSs is typically % of the stellar luminosity, but it can be comparable with or higher than the stellar luminosity for a few CTTSs (see [85]). WTTSs also emit at short wavelengths, from X-rays to UV, but their excess luminosity at these wavelengths is comparable with or slightly higher than that of active main-sequence stars [2]. Similar to these, WTTS excesses are thought to be powered by magnetic activity at the stellar surface; however, magnetic activity cannot produce a luminosity comparable to that of the star, as seen in some CTTSs. A source of energy external to the star is required; this is the main justification for expecting CTTSs to be accreting matter from their disks and in the process releasing gravitational potential energy that powers the excess _ch2_P.tex /28/2 6:8 page 8

6 PROTOPLANETARY DISKS / 9 Broad emission lines [42, 36] and veiling of absorption lines and bands in the blue and ultraviolet regions of the spectra [98] have been detected in YBDs, and as in CTTSs, the source of the excess has been attributed to accretion energy. It is more difficult to measure veiling in HAeBe; however, measurements of an excess of flux in the Balmer discontinuity have also been interpreted as evidence of an excess of energy produced probably by accretion [35]. Similarly, association with jets and UV excess have been interpreted as due to accretion in these objects [74]. In the near IR, CTTSs show characteristic excesses as well. In particular, in the J-H versus H-K diagram, most CTTSs fall in a well-defined region, the CTTS locus (left panel of Fig. 2.2; [22]), while WTTSs have colors consistent with dwarf stars. The HAeBe also show characteristic colors in this diagram [97], which helps distinguish them from classical Be stars. Around μm, HAeBe stars show a characteristic bump in their SEDs [99], and it has been found that CTTSs also show a similar kind of excess, although it is less apparent because of the relatively larger photospheric emission at those bands [34]. This emission has been interpreted as produced by the inner wall of the disk made by gas and dust, located at the dust-sublimation radius [39, 7]. With near-infrared photometry, the inner radii have been measured for CTTSs and HAeBe stars [25] and have been found to be consistent with the radius expected for dust sublimation ( 4.3). Observations from the Infrared Space Observatory (ISO) telescope in space provided a wealth of information on the IR spectra of YSOs. Observations from the instruments on board Spitzer have now characterized the excesses Fig Left panel: Taurus stars in the J-H versus H-K diagram. The dwarf sequence and the CTTS locus (solid gray line) are shown. Observations have not been corrected for reddening. Right panel: Taurus stars in the IRAC [3.6] [4.5] versus [5.8] [8] diagram. The dotted square indicates the region covered by colors of irradiated accretion-disk models. Data from [82] _ch2_P.tex /28/2 6:8 page 9

7 2 / NURIA CALVET AND PAOLA D ALESSIO of TTSs, HAeBe and YBDs in these wavelength regions [66, 67, 62, 2, 2, 9, 28]. For instance, in color-color diagrams constructed with combinations of the four bands of the Infrared Array Camera (IRAC) instrument, at 3.6, 4.5, 5.8, and 8 μm, and the Multiband Imaging Photometer (MIPS) 24μm band, most CTTSs fall in a region well separated from the WTTSs. The right panel of Fig. 2.2 shows one of these diagrams. The WTTSs have colors, while most CTTSs populate a distinct region of the [3.6] [4.5] versus [5.8] [8.] diagram, which corresponds to the emission expected from optically thick disks [5, 82] ( 4). In addition, spectra of CTTSs between 5 and 3 μm obtained with the Infrared Spectrograph (IRS) on Spitzer have shown the large diversity of these spectra, including fluxes and profiles of the silicate features and the slope of the SED in this wavelength region [66, 2, 9, 28]. As discussed in 5, these observations give direct information on the spacial distribution and evolutionary state of the solid component in the disks. Another important observational development of the last years has been interferometric observations in the near and mid-ir of YSOs; with resolutions of a few milliarcseconds, these observations are probing the structure of the innermost disk regions and providing invaluable information about their structure [25]. Optical and near-ir scattered light disk images, combined with detailed modeling of the transfer of stellar radiation through the disk dust, have been an important tool to understand the disk geometry and to constrain the properties of its atmospheric dust [87]. In addition, observations with single dish and interferometers in the submillimeter and millimeter range are probing the midplane regions of the disks from AU and beyond [9, 7] With the Herschel Observatory, which is observing the region between 5 and 6 μm, we will have complete wavelength coverage of the SEDs of YSOs against which we should confront our models. As a summary of the emission properties of YSOs, Fig. 2.3 shows the SED of the CTTS BP Tau. The excess over the photospheric fluxes, shown with dashed lines, is clearly apparent from the UV to the mm. Shown are the observed fluxes and fluxes corrected for reddening. With the wealth of information of the last years, we have learned much about disk structure, for example, what the main disk-heating mechanisms are, the role of the dust, and how matter is accreted by the central star. There are still debates on the details of the accretion mechanisms and how to quantify the viscosity. We have also learned about disk evolution. We now know that mass-accretion rate and disk emission decrease with age, as expected in general terms from evolutionary models of the gas [86] and the solids [77, 55] in the disk. However, many points in this evolution from the primordial gas-rich _ch2_P.tex /28/2 6:8 page 2

8 PROTOPLANETARY DISKS / 2 Fig Spectral energy distribution of the CTTS BP Tau as an illustration of the typical SED of an accreting late-type star. The photosphere is shown with a dashed line. In the near IR, the solid circles correspond to the observations and the open circles to the data corrected for reddening. This correction is not important beyond the near IR for most nonembedded objects. The HST/ Space Telescope Imaging Spectrograph (STIS) UV data are from [9]; the optical photometry, Infrared Astronomy Satellite (IRAS) fluxes, and spectral type are from KH95; the IRS spectrum is from [66]; the millimeter data are from [7]. disks to the debris disks with planets are yet to be understood. In this contribution, we will outline some of the basic physical principles characterizing disk structure and emission, hoping to give the reader insight to understand and perhaps help clarify some of the many remaining problems. Although we will concentrate on CTTSs, the principles outlined are general enough to be applicable to stars in other mass ranges. 3. Magnetospheric Accretion and Mass-Accretion Rate The collapse of slowly rotating molecular cores forms stars surrounded by disks by conservation of angular momentum [66]. Most of the mass of the cores resides at large distances, which have the highest angular momentum, so it falls onto the disk and not onto the star. This matter has to be transported through the disk to the center and onto the star to make up the bulk of the stellar mass. One can define the disk mass-accretion rate Ṁ onto the star as the mass per year going through a cylindrical control surface centered in the star. The buildup of the stellar mass occurs mostly in the Class I phase, while the protostar is still embedded in its envelope and receiving matter from it; it is thought to occur in episodes of high mass accretion due to disk instabilities [85], which have been identified with the the outbursts in FU Ori objects [88, 9] _ch2_P.tex /28/2 6:8 page 2

9 22 / NURIA CALVET AND PAOLA D ALESSIO The YSOs we have discussed in 2 are at a later evolutionary phase, when most of the mass of the star has already been built. Still, as we have seen, the total luminosity of some of these objects exceeds that expected from release of magnetic energy on the stellar surface or even from the expected quasi-static contraction of a star in the pre-main-sequence phase, implying the existence of an external source of energy. For stars surrounded by disks, which is the case for the objects with large excess, a readily available energy source is gravitational potential energy if the disks are accreting matter onto the star. But can we confirm that this is the case, and if so, how is potential energy actually released? In standard steady accretion disks, matter is transfered from the disk to the star through a narrow boundary layer in which /2 of the accretion luminosity L acc = GM Ṁ/R is released as material slows down from the Keplerian velocity at R to the much lower stellar rotational velocity [85]. Although this model could explain in general terms the continuum excess [6, 5], other crucial observations could not be understood with it. As noted, CTTSs have strong emission lines; the peaks of these lines are close to the line center, and their wings extend to hundreds of km s [8, 3, 3, 33]. These emission lines exhibit blueshifted absorption components, usually attributed to the matter ejected from the star, so early models sought to explain the emission lines as formed in winds [8, 4]. However, the profile of emission lines in CTTSs are different from those produced in a spherically symmetric wind for expected wind-velocity profiles. In particular, they are nearly centrally peaked, and, moreover, a redshifted absorption component with characteristic velocities of km s is seen in some cases [63], in addition to the blueshifted component. These redshifted absorptions, indicative of high-velocity infalling material, which coexists with outflowing material, cannot be understood with the standard accretion-disk model. The present-day paradigm for transferring matter from the disk to the star is magnetospheric accretion. In this model, the stellar magnetic field truncates the inner disk, and matter falls onto the star along magnetic-field lines, merging with the photosphere through an accretion shock at the stellar surface. Simultaneously, matter is ejected through open field lines, most likely in the innermost regions of the disk. The virtue of this model is that it can consistently explain a number of observations. To start with, disks are expected to be truncated at a few stellar radii, given the typical mass-accretion rate and the strength of the stellar magnetic field. In spherical infall onto a magnetized body, if the gas is sufficiently ionized, matter cannot move freely inside a given distance r, where the infall velocity and density are v and ρ, respectively, _ch2_P.tex /28/2 6:8 page 22

10 PROTOPLANETARY DISKS / 23 such that B 2 /8π >/2 ρv 2 ; rather, matter couples to the magnetic field B, and accretion may even be stopped [65]. The radius where the magnetic pressure equals the ram pressure is r ( i B ) 4/7 ( Ṁ ) 2/7( M ) /7( R ) 5/7 = 7. R KG 8 M yr.5 M 2R In accretion disks, the truncation radius is a fraction of this value, /3 2/3 [65]. Surface magnetic-field strengths in TTSs are of the order of KGs [9], and the average mass-accretion rate is of the order of 8 M yr [89, 8], so disks should be truncated at a few stellar radii. Observed emission-line profiles are naturally explained if these lines form in the magnetic infall region, as shown schematically in Fig The bulk of the line forms in the region where matter is just lifted from the disk; this matter has very low velocities but large emitting volumes. In contrast, the wings of the line form near the star where matter is approaching the surface at the free-fall velocities of a few kms. Moreover, this high-velocity matter may absorb the background accretion-shock emission for appropriate line-of-sight inclinations, which naturally explains the redshifted absorption component. Detailed models confirm these expectations [3, 33]. Figure 2.5 shows observed profiles of Hα and Na I 5876 for three CTTSs in increasing order of mass-accretion rate from bottom to top (determined from their UV excess and veiling); observations are compared with the magnetosphericmodel predictions. The comparison indicates that emission lines form in the magnetospheric infall flows for all but the highest-accretion-rate CTTSs. For the high accretors, high-opacity lines like Hα have a typical wind profile, but lower-opacity lines like Na I D are magnetospheric in nature [33]. Fig Schematic of line formation in magnetospheric flow. Adapted from [23] _ch2_P.tex /28/2 6:8 page 23

11 24 / NURIA CALVET AND PAOLA D ALESSIO Fig Observed line profiles (solid lines) compared with institutions of the magnetospheric accretion model (dashed lines). Stars have increasing mass-accretion rates in their disks from bottom to top, determined from veiling measurements. The magnetospheric accretion model explains fairly well the observed profiles except in high-opacity lines like Hα in high accretors like DR Tau (upper left panel), for which the line is formed mainly in the wind [4]. Even in those cases, the profiles of lower-opacity lines like Na D (upper right panel) indicate that those lines form in the magnetosphere. Adapted from [33]. Finally, the magnetospheric accretion model can explain the flux excess observed in the optical and UV wavelengths in CTTSs as formed in the accretion shock at the stellar surface. Matter approaches the photosphere at free-fall velocities and forms an accretion shock very near the surface, where matter heats to temperatures 2 T s = ( M.5 M )( R 2R ) K _ch2_P.tex /28/2 6:8 page 24

12 PROTOPLANETARY DISKS / 25 Fig Schematic of accretion column. Matter falling at free-fall velocities reaches the stellar surface, with which it merges through an accretion shock. Kinetic energy thermalizes, and for the expected temperatures, the shock emits soft X-rays, which heat the preshock region and the photosphere below the shock. These regions reprocess the shock emission and emit mostly in the UV and optical wavelengths [26]. Soft X-ray radiation from the shock heats the preshock region and the photosphere just below the shock, as indicated schematically in Fig. 2.6; the reprocessed emission from these regions adds to the photospheric emission as an excess continuum [26]. Plate 2 shows that the predicted emission from the accretion-shock model fits fairly well the HST and ground-based observations of a typical CTTS. Recent modeling of the X-ray spectra of one CTTS, TW Hya, also shows that the accretion-shock model can explain the features in at least this star [62], although this may not be the case for all CTTSs [69]. The models discussed above are very simple in the sense that they are axially symmetric, assuming homogeneous flows. Reality, of course, is much more complicated, as the variability observed in excess continuum and emission lines readily shows. Magnetic field and rotation axis are most likely not aligned, and flows are not uniform sheets onto the star. But the main physical principles are present in these models. In particular, the energetic budget is modified from that of the standard model, in which the excess continuum could have at most half of the accretion luminosity, dissipated in the boundary layer, while the other half was emitted by the disk. In contrast, in the magnetospheric model the intrinsic disk emission is drastically reduced because matter is being lifted from it at a few stellar radii, depriving the disk of the contributions of regions deep in the stellar potential well. This potential energy, rather, comes out in the magnetospheric infalling material. This is fortunate because the luminosity in the excess continuum is then of the order of the accretion _ch2_P.tex /28/2 6:8 page 25

13 26 / NURIA CALVET AND PAOLA D ALESSIO Fig Mass-accretion rates for stars of a number of populations indicated in the insert with ages between and Myr. These include stellar clusters and distributed populations. From [3]. luminosity L acc ; for known M and R, given by the position of the star in the HR diagram, measurements of the excess luminosity allow us to estimate the mass-accretion rate onto the star, an important quantity for understanding disk structure. On the basis of these principles, measurements of the excess luminosity in the 3,2 5,3 Å range and conversion to L acc have yielded mass-accretion rates for a sample of 8 stars [75]. These measurements have been used to establish secondary calibrations of the accretion luminosity in terms of observables easier to obtain, such as the excess luminosity in the U band [75] and the luminosity in Ca II 8542, Paschen β, and Brγ lines [3, 29, 22, 28]. These calibrations have been used to determine accretion luminosities and mass-accretion rates in numerous samples of stars [86, 8]. The average mass-accretion rate in CTTSs belonging to Myr old populations is 8 M yr. Moreover, determinations of mass-accretion rates for populations in the Myr age range using these methods indicate that the accretion rate decreases with age roughly as age.5, as shown in Fig. 2.7, in agreement with predictions of viscous evolution [86]. One of the main caveats of this procedure is the so-called bolometric correction, that is, the conversion from the excess luminosity measured in the observing wavelength range to the total excess luminosity, which relies on the model used. Ref. [75] and others adopted a 4 K slab model with variable optical depth. Using an accretion-shock model, [26] found corrections that were consistent with those of [75]; this agreement is due to the fact that the emission from a hot slab with optical depth is not very different _ch2_P.tex /28/2 6:8 page 26

14 PROTOPLANETARY DISKS / 27 from the emission of the shock, which essentially consists of the sum of the blackbody-like emission from the photosphere below the shock, which reaches similarly hot temperatures, and a smaller contribution from the optically thin pre shock region. In any case, recent determinations [8, 64] of the wavelength dependence of the veiling have shown that the excess flux in the 5,, Å range is higher than predicted by the single accretion column of Ref. [26]. A more realistic model, including a diversity of accretion columns carrying different energy fluxes, can better explain the veiling observations; the excess at longer wavelengths could be due to accretion columns carrying low-energy fluxes and thus heating the photosphere to lower temperatures than the highenergy columns that contribute to the UV [3]. The resultant mass-accretion rates are higher by a factor of 2 to 3, which is within the uncertainties of the determinations [3]. The determination of the mass-accretion rate from the veiling excess or by direct measurements of the UV excess continuum is limited by the intrinsic chromospheric emission of the star. WTTSs show levels of magnetic activity comparable to those of the most active stars. This shows in emission lines as Ca II triplet [2] and in their X-ray luminosities [6]. CTTSs are in the same evolutionary stage, and thus their chromospheres and transition regions must be similar. Therefore, emission from accretion flows with mass-accretion rates much lower than the average are not detectable in the continuum. However, accretion can be detected in high-opacity emission lines as Hα by the presence of high-velocity wings sometimes superimposed on the narrow chromospheric emission component. Thus Ref. [79] proposed that the % half width of Hα is a much better discriminant between CTTSs and WTTSs than the usually used Hα equivalent width (see 2). Similarly, modeling of the emission lines produced in the low-density flows of extremely low accretors, which are not affected by saturation effects, is the best way to estimate the mass-accretion rates. Such models have been done for the Hα and Br γ lines of very low-mass stars and YBDs [42, 36] and have yielded mass-accretion rates on the order of 2 M yr. The situation is also more complicated for HAeBe stars. In this case, the intrinsic photospheric emission is much higher than the excess; in addition, the photospheric emission peaks at wavelengths similar to those of the expected shock emission because both photosphere and shock have similar temperatures. These factors make the intrinsic shock emission much more difficult to detect, except by a small filling in of the Balmer jump [35]. Emission-line profiles of at least the HAeBe (least likely to be confused with classical Be stars) seem to be consistent with magnetospheric infall [35], but _ch2_P.tex /28/2 6:8 page 27

15 28 / NURIA CALVET AND PAOLA D ALESSIO the contribution from the wind seems to be much more important than in the TTS case. Secondary calibrations of L acc versus L(Br γ ), determined from measurements of mass accretion in intermediate mass CTTSs, predecessors of the HAeBe and still cool enough that the excess can be measured in the UV [28], are commonly used [69], but justification for their application from first principles is still to be given. In any event, measurements of the accretion luminosity now exist from the substellar limit to the intermediate-mass range. These measurements indicate a dependence Ṁ M 2 [36]. Several models have been proposed to explain this dependence [83, 53, 37], but none has yet been widely accepted. 4. Disks in YSOs The first images of disks around CTTSs came from the Very Lange Array (VLA) at 7 mm [54, 85] and from HST observations [2, 45, 46] Observations by millimeter interferometers have now imaged a substantial number of disks [9, 7], from which we are getting more and more detailed information on their velocity field, which is usually Keplerian, and on their surface-density distribution. Several groups now have codes for sophisticated calculations of disk structure and emission (see [52]). Here we focus on the general physical principles governing the structure and emission of disks around young stars and use our models to illustrate the results. 4.. Irradiated Accretion Disks Even though disks around YSOs are accreting (see 3), their SEDs do not agree with the predictions of standard accretion disks; specifically, they are flatter than λf λ λ 3/4 [5, 85]. However, for an average mass-accretion rate of 8 M yr ( 3) and typical stellar parameters, L acc. L. L, the stellar luminosity is higher than the accretion luminosity in most cases, indicating that stellar irradiation must be an important heating agent. For a flat disk, the irradiation flux can be estimated as 3 F irr I Ω cos θ I πr 2 R 2 h R 2I 3 ( R R where θ is the angle between the line connecting from a representative point on the stellar surface (at a height h 2R /3π above the disk midplane) and a point on the disk surface at distance R and the vector normal to this surface. Here we have approximated the stellar radiation as impinging on the disk surface in a single beam. If we take F irr σ T 4, the temperature is T R 3/4, ) 3, _ch2_P.tex /28/2 6:8 page 28

16 PROTOPLANETARY DISKS / 29 which is similar to the temperature distribution of an accretion disk, and thus the flat irradiated disk results in SEDs that cannot explain the observations either. Kenyon & Hartmann (987) [5] proposed that disks of CTTSs are not flat but flared, which means that they can capture more stellar radiation than flat disks. The equation of hydrostatic equilibrium in the vertical direction for a geometrically thin disk in the central gravitational potential well of the star can be written as dp 4 ρ dz = GM z R 3, where ρ(z, R) and p(z, R) are the mass density and pressure at height z and radius R. If the disk is isothermal in the vertical direction, then p = ρc 2 s, where c s = (kt/m) /2 is the sound speed, which does not depend on z, and m is the mean mass of the gas particles. With this approximation 5 ρ(z, R) = ρ m e z2 2H 2, where ρ m (R) is the density at the midplane and H is the gas scale height, given by c s 6 H = (GM /R 3 ) /2 = c s T /2 R 3/2. Ω K If temperature decreases with distance more slowly than R 3, then the disk gas scale height increases with distance. Then, if the height of the disk surface that intercepts the stellar radiation is proportional to the gas scale height, it is curved, capturing more stellar flux than a flat disk. With these approximations, T R 3/7, and the corresponding SEDs are much flatter than a viscous disk or a flat irradiated disk, i.e., they have a larger excess at longer wavelengths. Disks are not vertically isothermal. Stellar radiation enters the disk at an angle θ to the local normal to the stellar surface (again approximating the stellar radiation as coming in a single beam), so the energy flux captured by the disk is (σ T 4 )(R /R) 2 μ, with μ = cos θ [24, 25, 2, 33, 43, 44, 49]. For simplicity, the radiation field can be separated into two frequency ranges, the stellar range, given by the stellar energy distribution, which is related to the stellar effective temperature, and the disk range, given by the local emissivity and related to the disk local temperature. As stellar flux enters the disk, a fraction dτ /μ is absorbed at each z, where τ is the mean optical depth at the stellar range. This energy reemerges at the wavelength characterizing the local temperature, the disk range, so the direct impinging stellar flux decreases with height. The main opacity source in CTTS disks is dust grains, because temperatures are low enough that dust is not sublimated in most of the disk, _ch2_P.tex /28/2 6:8 page 29

17 3 / NURIA CALVET AND PAOLA D ALESSIO except in regions very close to the central star. Dust opacity increases as λ decreases, and stellar radiation is emitted at a shorter wavelength than the radiation emitted by the disk, so the hotter the star, the larger the opacity at the stellar range, and the higher the energy capture and thus the heating. The vertical temperature profile can be obtained from the equation of conservation of energy, 7 ρκ ν B ν (T)dν ρκ ν J d dν = df d 4π dz, where κ ν is the monochromatic absorption coefficient per unit of (gas plus dust) mass, J d is the mean intensity of the local radiation at the disk wavelength range, and F d is the local radiative flux. This equation indicates that the emitted energy is due to the absorption of photons from the radiation field plus the change in local flux F d. If we neglect viscous heating, the local flux changes only by deposition of stellar energy, so df d 8 dz 4πκ ρj = 4πκ ρj, e τ /μ, where κ is the mean opacity at the stellar wavelength range, and J, is the mean stellar intensity at the disk surface, given by J, = /4π IdΩ I Ω /4π. If we perform the integrals, equation (7) can be written as 9 κ P (T) σ T 4 (z) π = κ P (T)J d (z) + κ P (T ) σ T 4 4π ( R R ) 2 e τ /μ, where κ κ P (T ), and κ P is the Planck mean opacity ([49, 44]). In the surface, where the local field is much smaller than the stellar field, J d << J, ; in addition, the medium is optically thin, so τ /μ <<, and we can write ( ) 2 R. κ P (T )T 4 (z) κ P(T )T 4 This is an implicit equation for the surface temperature T, corresponding to the optically thin limit. Note that this is the hot-layer temperature in the 2-layer approximation [33]. Stellar heating and thus the temperature decrease as radiation penetrates the disk, because the optical depth increases (cf. eq. (9)). The actual T profile depends on μ, and through this, on the actual shape of the surface of the disk, defined as the surface where τ /μ, that is, where most of the stellar energy is deposited. The mass surface density of the upper optically thin region above the surface is given by ΔΣ μ /κ ; the more flared the 2 R _ch2_P.tex /28/2 6:8 page 3

18 PROTOPLANETARY DISKS / 3 disk surface, the larger the μ and the higher the mass of the optically thin region [4]. The stellar flux intersected by the disk surface z s (R) can be written as [5] [ F irr (R, z s ) σ T 4 ( ) 2 3 ( ) ] 2 R R Rd(z s /R) + π π 3 R R dr for R >> R. A flat disk has a negligible d(z s /R)/dR; thus 2 F flat = 2σ T ( R ) 3, irr 3π R and T flat (F flat irr /σ )/4 R 3/4, as we already have shown. For a flared disk, if we assume that the height of the surface is a fixed number of scale heights, z s becomes a function of T, and 3 Rd(z s /R) dr = z s dt 2 T dr 5 z s 2 R. With F irr = σ T 4, eqs. () and (3) form a system for T(R) in the isothermal approximation, resulting in T(R) R 3/7. If T increases, the gas scale height increases, then the cross section to capture stellar photons increases, and the disk heating increases. This T(R) R 3/7 solution has been found to be stable under perturbations in temperature or scale height as long as the thermal timescale of the disk is longer than the vertical dynamic timescale [43]. However, if the opposite is true, as could be the case for the outer disk, this simple solution turns out to be unstable [57]. The simple flared-disk solution is based on several simplifying assumptions. Effects like viscous dissipation, radial energy transfer, and scattering of the incident radiation tend to stabilize the disk. Moreover, disks are not isothermal, and the height of the surface is not a fixed number of scale heights. Results of the detailed solution of the set of equations of vertical structure, including viscous dissipation for a disk with typical CTTS parameters are shown in Figs. 2.8 and 2. [44]. Figures 2.8 and 2.9 shows the height of the surface z s, the scale height H, and the photospheric height z phot, where the Rosseland mean optical depth τ Ross. Note that the height z phot is defined only in the region where the disk is optically thick to its own radiation, determined by the Rosseland mean opacity. In the example shown in Figs. 2.8 and 2.9, the disk becomes optically thin to its own radiation (τ Ross < ) for R > 2 AU. In contrast, since the opacity at the wavelength where the stellar radiation is absorbed (λ μm for T 4K) is large, the disk remains optically thick to the stellar radiation, and the surface is flared out to a few hundred AU _ch2_P.tex /28/2 6:8 page 3

19 32 / NURIA CALVET AND PAOLA D ALESSIO Fig Characteristic heights in the disk. Height z s where stellar radiation is absorbed (solid line); scale height H m (dashed line); photospheric height z phot (circles), defined where the disk is optically thin to its own radiation. Model parameters are M =.5 M, R = 2R, T = 4, K and Ṁ = 8 M yr. Dust is uniformly mixed with gas and has an ISM composition and size distribution. Adapted from [44, 48]. Fig Same as Fig. 2.8, but for grains with a size distribution characterized by a max = mm. Figures 2. and 2. shows characteristic temperatures in the disk. It can be seen that the midplane temperature T m is higher than the photospheric temperature T phot = T(z phot ) in the inner disk, where it is optically thick to its own radiation. This can be understood by using the diffusion approximation 4 dj d dz = σ dt 4 π dz = 3 4π χ RρF d, _ch2_P.tex /28/2 6:8 page 32

20 PROTOPLANETARY DISKS / 33 Fig. 2.. Characteristic temperatures of the disk. Midplane temperature T m (solid line); upperlayers temperature T (dotted line); photospheric temperature T phot (circles), defined where the disk is optically thin to its own radiation; viscous temperature T vis (dashed line). Model parameters are M =.5 M, R = 2R, T = 4, K, and Ṁ = 8 M yr. Dust is uniformly mixed with gas and has an ISM composition and size distribution. Adapted from [44, 48]. Fig. 2.. Same as Figure 2., but for grains with a size distribution characterized by a max = mm. from which we can write 5 Δ(σ T 4 ) 3 4 τ RF d. In the inner annuli, τ R >>, and T m > T phot ; the temperature gradient allows the flux (viscous plus local radiation) to emerge from the disk. In the _ch2_P.tex /28/2 6:8 page 33

21 34 / NURIA CALVET AND PAOLA D ALESSIO outer regions, when the disk becomes optically thin to its own radiation, τ R <<, the disk becomes nearly isothermal in the regions near the midplane. As shown in Figs. 2. and 2., the midplane temperature remains at the dust-destruction temperature (,5 K) in the innermost regions; if the temperature increases, the dust gets destroyed and the opacity drops; as a result, the disk becomes optically thin and cools below the dust-destruction temperature, at which point the opacity increases again. This thermostat effect makes the temperature at the midplane stay at the dust-destruction temperature [89]. The surface temperature T is higher than T phot in the inner regions and than T m in the outer optically thin regions, as predicted by eq. (9). By comparison of T vis and T phot in Figs. 2. and 2., it can be seen that viscous heating is important only in regions inside AU, given the low Ṁ characteristic of the typical CTTS. Finally, it can be seen that temperatures behave as /R /2 for R >> R. As a summary, the upper panel of Fig. 2.2 shows isocontours of temperature for the disk model in Figs. 2.8 and 2.. It also shows isocontours of number density for the same cases, and the surface z s where stellar radiation is absorbed. The particular shape of the temperature profile has important observational implications. For one thing, features formed in the optically thin upper regions will appear in emission, even if the disk is optically thick, because the local temperature is so much higher than that of deeper regions where the continuum forms. As a result of this chromospheric effect, features like those from silicates that form in the atmosphere of the optically thick inner disk regions appear in emission [4, 66]. Molecular features formed in the upper layers appear in emission as the CO near infrared [24, 2] and water lines [3]. The higher temperatures of the upper layers also imply that molecules can exist in the gas phase in the disk even when the midplane temperatures are so low that molecules are settled onto grain surfaces [88, 9, 3, 2]. For example, if the disk shown in Fig. 2. were isothermal at T m, the CO molecules would be on grain surfaces for R > 6 AU, for which T < 2 K (see also Fig. 2.2). However, the hot upper layers are kept at a temperature high enough for molecules to be in the gas phase out to 4 AU, in agreement with millimeter molecular observations [59, 6, 49, 2]. An additional important effect further affects molecular equilibrium and emission. The results discussed so far assume that the temperatures of the gas and dust are the same. However, in the low-density uppermost layers of the disk, the collisional coupling between gas and dust becomes much less effective, and other factors become more important in heating and cooling the gas. Heating factors include absorption of X-rays and UV radiation, grain _ch2_P.tex /28/2 6:8 page 34

22 PROTOPLANETARY DISKS / 35 Fig Isocontours of temperature and number density in the disk. Isocontours are shown for temperatures (solid lines) 8 K, 7 K, 6 K, 5 K, 4 K, 3 K, and 2 K, and densities (dashed lines) 6, 7, 8 and 9 cm 3. The height where most of the energy in the stellar radiation is absorbed is indicated by a dotted line. The disk model parameters are M =.5 M, R = 2R, T = 4, K, and Ṁ = 8 M yr. Upper panel: ISM dust. Lower panel: a max = mm. Adapted from [44, 48]. photoelectric heating, exothermic chemical reactions, and collisions with warm grains, while cooling factors include line emission and collisions with cooler dust grains. Several groups are working on the calculation of gas temperatures [7, 7, 72,, 73, 44], and not all groups include all these factors. The gas temperature in the low-density, uppermost layers of the disk becomes much higher than the dust temperature, reaching 5, K at AU and 3 K at AU. These elevated temperatures imply that molecules can be in the gas phase even in the uppermost layers of the outer disk, in agreement with observations. The surface density of the disk can be self-consistently calculated from the equations of an irradiated accretion disk. A geometrically thin accretion disk with a steady mass-accretion rate Ṁ has a surface-density distribution given by the conservation of the angular momentum flux, _ch2_P.tex /28/2 6:8 page 35

23 36 / NURIA CALVET AND PAOLA D ALESSIO [ 6 Σ = Ṁ ( ) ] /2 R, 3πν R where ν is the viscosity. In the parametric α prescription [57], the viscosity can be written as ν = αc s H = αcs 2/Ω K by using eq. ( 6). With c s T /2 R /4 and Ω K M /2 R 3/2, we obtain at large radii 7 Σ 4 Ṁ 8 M yr ( α. ) ( ) ( ) T AU R ( ) /2 M gr cm 2, K AU M using values of α found in modeling CTTS disks and expected from theories (see the chapter Balbus, this volume), and typical temperatures at AU. The surface-density dependence on radius of irradiated accretion disks, Σ R, is much flatter then the usually assumed dependence Σ R.5, and it has been confirmed by observations [84, 8]. If we assume that this dependence holds at all radii, the disk mass would be given by 8 M d M =.3 Ṁ 8 M yr ( ) Rd ( α ) AU. ( ) ( ) /2 T AU M, K M in agreement with values determined from dust millimeter emission. We can see that the mass-accretion rates, determined from the inner disk properties, are consistent with large-scale properties like the disk mass. Note that if Ṁ is known for a given star, then M d and α are complementary parameters, since the temperature at the outer disk radii, T Rd, is fixed by stellar irradiation, and sizes can be estimated from observations Effects of Dust Properties The temperature of a volume element inside the disk, assumed to be in thermal equilibrium (i.e., such that its temperature does not change with time), is mostly controlled by the balance between the heating produced through absorption of stellar radiation, accretion-shock radiation, and the fraction of disk radiation that reaches the volume element( 4.), and the cooling due to the radiative losses of the element. There are different mechanisms that transport energy between disk regions, contributing to local heating _ch2_P.tex /28/2 6:8 page 36

24 PROTOPLANETARY DISKS / 37 and cooling, but radiation is the most relevant of these mechanisms [49]. Another important aspect to keep in mind is that the disk temperature is not an isolated quantity but is connected to the disk viscosity, gas scale height, density, and other factors, so the whole disk structure is an interdependent phenomenon. The crucial ingredient in the absorption and emission of radiation is the opacity of the disk material, and given the low temperatures in almost the whole disk (around a young low- or intermediate-mass star), dust happens to be its main opacity source, despite the fact that it represents only around % of the disk mass. In this subsection we summarize important dust properties and the general effects they have on the structure and emission of circumstellar disks. The dust opacity depends on the shape, size distribution, abundance, and constitution of the dust grains. Schematically, a spherical grain with radius a has a cross section for absorption of radiation at wavelength λ of the order of the geometric cross section πa 2,ifλ<<2πa, and it decreases as λ 2 for λ>>2πa. It is common to use the size parameter x = λ/2πa and an efficiency factor Q a, defined as the cross section for absorption divided by the geometric cross section. Thus Q a for x <, and Q a x 2 for x >. The dust opacity, κ ν (a), given in cm 2 per gram of dust, is given by the cross section over the mass of the grain, m g = ρ g 4/3πa 3, where ρ g is the bulk density of the grain. Thus κ ν /a, which means that the bigger the grain, the lower the opacity at short wavelengths and the higher at long wavelengths, close to where x. Note that the transition to the λ 2 regime occurs at a wavelength that increases with a. This is shown schematically in Fig In general, dust grains are not of a single size, and typically there are different numbers of grains in different size intervals, between a minimum radius a min and a maximum radius a max. This is described by a size-distribution function, for example, a power law n(a)da a p da, where a is the grain size and the exponent p is usually taken as 3.5, describing the properties of the ISM dust [5], or 2.5 if there has been some degree of coagulation [26]. The coefficient in the dust size distribution is proportional to the dust-to-gas mass ratio, ζ, which specifies the mass in dust in a given disk mass element. If a min and ζ are fixed, the larger the a max, the less the number of smaller particles, because the larger particles take more mass, and therefore, the lower the opacity at short wavelengths and the higher at long wavelengths. For a mixture of sizes, the transition from the flat to the λ 2 regime occurs over a large range of wavelengths (see Fig. 2.3), so that the local slope of the function κ(λ) versus λ changes slowly from to 2. Therefore, the form usually assumed to represent the dust opacity κ λ β is not actually valid, since β depends on _ch2_P.tex /28/2 6:8 page 37